American Institute of Mathematical Sciences

Advanced
October  2014, 10(4): 1041-1058. doi: 10.3934/jimo.2014.10.1041

On a risk model with randomized dividend-decision times

Zhimin Zhang 1,

1. 

College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, China

Received  October 2012 Revised  December 2013 Published  February 2014

In this paper, we consider a perturbed compound Poisson risk model with a randomized dividend strategy. Assume that decisions on paying off dividends are made at some random observation times. Whenever the observed value of the surplus process exceeds a given barrier $b$, the excess value will be paid off as dividends. We assume that the Laplace transform of the individual claim size belongs to the rational family. When the time intervals between successive decision times follow exponential distribution, we present explicit expressions for the Gerber-Shiu function. We also extend the exponential assumption to Erlang and discuss the solution procedure.
Keywords: Erlang., integral equation, Gerber-Shiu function, Dividend-decision time, exponential.
Mathematics Subject Classification: Primary: 91B30; Secondary: 62P0.
Citation: Zhimin Zhang. On a risk model with randomized dividend-decision times. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1041-1058. doi: 10.3934/jimo.2014.10.1041
References:
[1]

H. Albrecher, E. C. K. Cheung and S. Thonhauser, Randomized observation periods for the compound Poisson risk model: Dividends,, Astin Bulletin, 41 (2011), 645.   Google Scholar

[2]

H. Albrecher, E. C. K. Cheung and S. Thonhauser, Randomized observation periods for the compound Poisson risk model: The discounted penalty function,, Scandinavian Actuarial Journal, 2013 (2013), 424.  doi: 10.1080/03461238.2011.624686.  Google Scholar

[3]

B. Avanzi, E. C. K. Cheung, B. Wong and J.-K. Woo, On a periodic dividend barrier strategy in the dual model with continuous monitoring of solvency,, Insurance: Mathematics and Economics, 52 (2013), 98.  doi: 10.1016/j.insmatheco.2012.10.008.  Google Scholar

[4]

B. De Finetti, Su un impostazione alternativa della teoria collectiva del rischio,, Transactions of the XVth International Congress of Actuaries, 2 (1957), 433.   Google Scholar

[5]

H. U. Gerber and E. S. W. Shiu, On the time value of ruin,, North American Actuarial Journal, 2 (1998), 48.  doi: 10.1080/10920277.1998.10595671.  Google Scholar

[6]

H. U. Gerber and E. S. W. Shiu, Optimal dividends: Analysis with Brownian motion,, North American Actuarial Journal, 8 (2004), 1.  doi: 10.1080/10920277.2004.10596125.  Google Scholar

[7]

A. E. Kyprianou, Introductory Lectures on Fluctuations of Lévy Processes with Applications,, Springer-Verlag, (2006).   Google Scholar

[8]

S. Li, The distribution of the dividend payments in the compound poisson risk model perturbed by diffusion,, Scandinavian Actuarial Journal, 2006 (2006), 73.  doi: 10.1080/03461230600589237.  Google Scholar

[9]

S. Li and J. Garrido, On ruin for the Erlang(n) risk model,, Insurance: Mathematics and Economics, 34 (2004), 391.  doi: 10.1016/j.insmatheco.2004.01.002.  Google Scholar

[10]

S. Li and J. Garrido, On a class of renewal risk model with a constant dividend barrier,, Insurance: Mathematics and Economics, 35 (2004), 691.  doi: 10.1016/j.insmatheco.2004.08.004.  Google Scholar

[11]

X. S. Lin, G. E. Willmot and S. Drekic, The classical risk model with a constant dividend barrier: Analysis of the Gerber-Shiu discounted penalty function,, Insurance: Mathematics and Economics, 33 (2003), 551.  doi: 10.1016/j.insmatheco.2003.08.004.  Google Scholar

[12]

X. S. Lin and K. P. Pavlova, The compound Poisson risk model with a threshold dividend strategy,, Insurance: Mathematics and Economics, 38 (2006), 57.  doi: 10.1016/j.insmatheco.2005.08.001.  Google Scholar

[13]

X. S. Lin and K. P. Sendova, The compound Poisson risk model with multiple thresholds,, Insurance: Mathematics and Economics, 42 (2008), 617.  doi: 10.1016/j.insmatheco.2007.06.008.  Google Scholar

[14]

C. C. L. Tsai and G. E. Willmot, A generalized defective renewal equation for the surplus process perturbed by diffusion,, Insurance: Mathematics and Economics, 30 (2002), 51.  doi: 10.1016/S0167-6687(01)00096-8.  Google Scholar

[15]

Z. Zhang and X. Wu, Dividend payments in the Brownian risk model with randomized decision times,, Acta Mathematicae Applicatae Sinica-English Series, (2013).   Google Scholar

show all references

References:
[1]

H. Albrecher, E. C. K. Cheung and S. Thonhauser, Randomized observation periods for the compound Poisson risk model: Dividends,, Astin Bulletin, 41 (2011), 645.   Google Scholar

[2]

H. Albrecher, E. C. K. Cheung and S. Thonhauser, Randomized observation periods for the compound Poisson risk model: The discounted penalty function,, Scandinavian Actuarial Journal, 2013 (2013), 424.  doi: 10.1080/03461238.2011.624686.  Google Scholar

[3]

B. Avanzi, E. C. K. Cheung, B. Wong and J.-K. Woo, On a periodic dividend barrier strategy in the dual model with continuous monitoring of solvency,, Insurance: Mathematics and Economics, 52 (2013), 98.  doi: 10.1016/j.insmatheco.2012.10.008.  Google Scholar

[4]

B. De Finetti, Su un impostazione alternativa della teoria collectiva del rischio,, Transactions of the XVth International Congress of Actuaries, 2 (1957), 433.   Google Scholar

[5]

H. U. Gerber and E. S. W. Shiu, On the time value of ruin,, North American Actuarial Journal, 2 (1998), 48.  doi: 10.1080/10920277.1998.10595671.  Google Scholar

[6]

H. U. Gerber and E. S. W. Shiu, Optimal dividends: Analysis with Brownian motion,, North American Actuarial Journal, 8 (2004), 1.  doi: 10.1080/10920277.2004.10596125.  Google Scholar

[7]

A. E. Kyprianou, Introductory Lectures on Fluctuations of Lévy Processes with Applications,, Springer-Verlag, (2006).   Google Scholar

[8]

S. Li, The distribution of the dividend payments in the compound poisson risk model perturbed by diffusion,, Scandinavian Actuarial Journal, 2006 (2006), 73.  doi: 10.1080/03461230600589237.  Google Scholar

[9]

S. Li and J. Garrido, On ruin for the Erlang(n) risk model,, Insurance: Mathematics and Economics, 34 (2004), 391.  doi: 10.1016/j.insmatheco.2004.01.002.  Google Scholar

[10]

S. Li and J. Garrido, On a class of renewal risk model with a constant dividend barrier,, Insurance: Mathematics and Economics, 35 (2004), 691.  doi: 10.1016/j.insmatheco.2004.08.004.  Google Scholar

[11]

X. S. Lin, G. E. Willmot and S. Drekic, The classical risk model with a constant dividend barrier: Analysis of the Gerber-Shiu discounted penalty function,, Insurance: Mathematics and Economics, 33 (2003), 551.  doi: 10.1016/j.insmatheco.2003.08.004.  Google Scholar

[12]

X. S. Lin and K. P. Pavlova, The compound Poisson risk model with a threshold dividend strategy,, Insurance: Mathematics and Economics, 38 (2006), 57.  doi: 10.1016/j.insmatheco.2005.08.001.  Google Scholar

[13]

X. S. Lin and K. P. Sendova, The compound Poisson risk model with multiple thresholds,, Insurance: Mathematics and Economics, 42 (2008), 617.  doi: 10.1016/j.insmatheco.2007.06.008.  Google Scholar

[14]

C. C. L. Tsai and G. E. Willmot, A generalized defective renewal equation for the surplus process perturbed by diffusion,, Insurance: Mathematics and Economics, 30 (2002), 51.  doi: 10.1016/S0167-6687(01)00096-8.  Google Scholar

[15]

Z. Zhang and X. Wu, Dividend payments in the Brownian risk model with randomized decision times,, Acta Mathematicae Applicatae Sinica-English Series, (2013).   Google Scholar

[1]

Yanbin Tang, Ming Wang. A remark on exponential stability of time-delayed Burgers equation. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 219-225. doi: 10.3934/dcdsb.2009.12.219
[2]

Serge Nicaise, Cristina Pignotti, Julie Valein. Exponential stability of the wave equation with boundary time-varying delay. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 693-722. doi: 10.3934/dcdss.2011.4.693
[3]

Y. T. Li, R. Wong. Integral and series representations of the dirac delta function. Communications on Pure & Applied Analysis, 2008, 7 (2) : 229-247. doi: 10.3934/cpaa.2008.7.229
[4]

Bai-Ni Guo, Feng Qi. Properties and applications of a function involving exponential functions. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1231-1249. doi: 10.3934/cpaa.2009.8.1231
[5]

Emad Attia, Marek Bodnar, Urszula Foryś. Angiogenesis model with Erlang distributed delays. Mathematical Biosciences & Engineering, 2017, 14 (1) : 1-15. doi: 10.3934/mbe.2017001
[6]

Wei Zhong, Yongxia Zhao, Ping Chen. Equilibrium periodic dividend strategies with non-exponential discounting for spectrally positive Lévy processes. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020087
[7]

Giovanni Colombo, Khai T. Nguyen. On the minimum time function around the origin. Mathematical Control & Related Fields, 2013, 3 (1) : 51-82. doi: 10.3934/mcrf.2013.3.51
[8]

Yaru Xie, Genqi Xu. Exponential stability of 1-d wave equation with the boundary time delay based on the interior control. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 557-579. doi: 10.3934/dcdss.2017028
[9]

Carlos Conca, Luis Friz, Jaime H. Ortega. Direct integral decomposition for periodic function spaces and application to Bloch waves. Networks & Heterogeneous Media, 2008, 3 (3) : 555-566. doi: 10.3934/nhm.2008.3.555
[10]

Wenxiong Chen, Congming Li, Biao Ou. Qualitative properties of solutions for an integral equation. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 347-354. doi: 10.3934/dcds.2005.12.347
[11]

Jeremy Levesley, Xinping Sun, Fahd Jarad, Alexander Kushpel. Interpolation of exponential-type functions on a uniform grid by shifts of a basis function. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020403
[12]

Josef Diblík, Zdeněk Svoboda. Asymptotic properties of delayed matrix exponential functions via Lambert function. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 123-144. doi: 10.3934/dcdsb.2018008
[13]

Elena Cordero, Fabio Nicola, Luigi Rodino. Time-frequency analysis of fourier integral operators. Communications on Pure & Applied Analysis, 2010, 9 (1) : 1-21. doi: 10.3934/cpaa.2010.9.1
[14]

Sergiu Aizicovici, Yimin Ding, N. S. Papageorgiou. Time dependent Volterra integral inclusions in Banach spaces. Discrete & Continuous Dynamical Systems - A, 1996, 2 (1) : 53-63. doi: 10.3934/dcds.1996.2.53
[15]

Minvydas Ragulskis, Zenonas Navickas. Hash function construction based on time average moiré. Discrete & Continuous Dynamical Systems - B, 2007, 8 (4) : 1007-1020. doi: 10.3934/dcdsb.2007.8.1007
[16]

Kei Matsuura, Mitsuharu Otani. Exponential attractors for a quasilinear parabolic equation. Conference Publications, 2007, 2007 (Special) : 713-720. doi: 10.3934/proc.2007.2007.713
[17]

Bin Dan, Huali Gao, Yang Zhang, Ru Liu, Songxuan Ma. Integrated order acceptance and scheduling decision making in product service supply chain with hard time windows constraints. Journal of Industrial & Management Optimization, 2018, 14 (1) : 165-182. doi: 10.3934/jimo.2017041
[18]

Xiaolong Han, Guozhen Lu. Regularity of solutions to an integral equation associated with Bessel potential. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1111-1119. doi: 10.3934/cpaa.2011.10.1111
[19]

Kazuki Himoto, Hideaki Matsunaga. The limits of solutions of a linear delay integral equation. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 3033-3048. doi: 10.3934/dcdsb.2020050
[20]

Phuong Le. Liouville theorems for an integral equation of Choquard type. Communications on Pure & Applied Analysis, 2020, 19 (2) : 771-783. doi: 10.3934/cpaa.2020036

2019 Impact Factor: 1.366

Tools

Metrics

  • PDF downloads (170)
  • HTML views (0)
  • Cited by (14)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences